tag:blogger.com,1999:blog-89853385010229872622017-08-10T02:14:25.521-05:00Pre-Cal 30S (Fall '06)An interactive learning ecology for students and parents in my Pre-Cal 30S class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.comBlogger178125tag:blogger.com,1999:blog-8985338501022987262.post-21817856333198905812007-02-05T14:19:00.000-06:002007-02-05T14:21:11.882-06:00The Adventure Continues ...Our adventures in blogging continue....<br /><br />Watch for new blogs going live February 5, 2007 ...<br /><br /><blockquote><ul><li><a href="http://am40sw07.blogspot.com">Applied Math 40S</a> (Winter '07) (Grade 12)</li><br /><li><a href="http://pc40sw07.blogspot.com">Pre-Cal 40S</a> (Winter '07) (Grade 12)</li><br /><li><a href="http://apcalc06.blogspot.com">AP Calculus AB</a> continues ... (Grade 12)</li></ul></blockquote><p>Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-69413645598555642472007-02-05T14:13:00.000-06:002007-02-05T14:18:23.779-06:00So Long ...<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://farm1.static.flickr.com/36/79811964_33754e8586_d.jpg"><img style="float:right; margin:0 0 10px 10px;cursor:pointer; cursor:hand;width: 320px;" src="http://farm1.static.flickr.com/36/79811964_33754e8586_d.jpg" border="0" alt="" /></a><br />And so we begin where we left off ... don't let the sky be your limit. ;-)<br /><br /><b><i>I'm so glad we've had this time together,<br /><br />Just to have a laugh or learn some math,<br /><br />Seems we've just got started and before you know it,<br /><br />Comes the time we have to say, "So Long!"</b></i><br /><br />So long everybody! Watch this space for pointers to new blogs for each of my classes. <br /><br />Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-16249172275437270802007-01-30T20:43:00.000-06:002007-01-30T21:16:59.583-06:00From Natnael .... To Pre-cal30sf06Hey guys,<br /><br />This is my last scribe post on this blog.<br /><br />I just wanna say a couple of things before i leave from Winnipeg. I really loved this class, and all of you guys. This was such an awesome experience for me, an dam sure for a lot of you. It was such a short time, and this class really kinda got us together on math. I've rarely done math with friends, i usually do it just individually, which i don't think i will continue doing, because of what i saw in the class.<br /><br />Oh about the postcard... guys ... really THANK YOU V.V.V.V.V.V.MUCH ... and Sam... i will try not to find friends better than you ;[]:...<br /><br />This class was such an amazing class i think. we all cooperated and worked together till the last minute (be4 the exam) Everyone was blogging, and sharing what he/she knew.. it was awesome.<br /><br />Well, guys, wish if i had staid with you guys longer.. at least till grad, but u know i have to move.. i guess if God wills we will get together sometime (theres a .009 chance..).<br /><br />Mr. K. you've been a wonderful teacher of the course, i really liked the way you would put examples on the board.. the pretest... the quizzes.. almost every thing.<br /><br />Hope you guys enjoy your upcoming academic years. AND EVERYONE IN THAT CLASS SHOULD STICK TO PRE-CAL, and i really do wish everyone good luck!!.. and am sure all of u will.<br /><br />Well its so sad, but.. this might be my last time talking to you guys...so good bye... in case anyone wanna add me to his/her msn this is ma email nathaniel_td@hotmail.com<br /><br /><br />Bye<br /><br />NatnaelNatnaelhttp://www.blogger.com/profile/04023224057112546632noreply@blogger.com4tag:blogger.com,1999:blog-8985338501022987262.post-65308079576174822752007-01-29T16:46:00.001-06:002007-01-29T16:46:51.639-06:00The year's come to an end ...<a href="http://bp0.blogger.com/_UT0W273SSqI/Rb55VbETbzI/AAAAAAAAACg/TGuaq_q5D0w/s1600-h/thankslani.jpg"><img id="BLOGGER_PHOTO_ID_5025587643142074162" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://bp0.blogger.com/_UT0W273SSqI/Rb55VbETbzI/AAAAAAAAACg/TGuaq_q5D0w/s400/thankslani.jpg" border="0" /></a><br /><div></div>cherrieeeeeeeehttp://www.blogger.com/profile/07176149235231149728noreply@blogger.com1tag:blogger.com,1999:blog-8985338501022987262.post-40503005626654165112007-01-28T17:53:00.000-06:002007-01-28T17:59:39.134-06:00Autograph that final exam and your life with EXCELLENCE!<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp3.blogger.com/_HiqPzpd3O9Y/RbZZyu9lVyI/AAAAAAAAAAM/7i9UVrobIB8/s1600-h/excellence.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp3.blogger.com/_HiqPzpd3O9Y/RbZZyu9lVyI/AAAAAAAAAAM/7i9UVrobIB8/s320/excellence.jpg" alt="" id="BLOGGER_PHOTO_ID_5023301162513028898" border="0" /></a><br />Hi there Pre-Cal 30s!<br /><br />Thank YOU so very much for those thank you posts!! You beat me to the punch! I wanted to be the first to thank YOU for allowing me to hang around and learn from you--<br /><br /><div style="text-align: center;"><span style="font-weight: bold;">AND</span></div><br />To wish you the very best!!<br />LaniLani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-54479304615047875922007-01-28T14:23:00.000-06:002007-01-28T14:28:07.353-06:00A BIG THANKS TO LANI!<span style="font-family:georgia;color:#663366;"><strong>Dear <em>Lani</em>,</strong> </span><br /><span style="font-family:georgia;color:#663366;">Since our semesters coming to an end, our class and myself personally would like to thank you for your contribution to our blog. You've definately added to our learning experience and your advice will be with us always. To think you took time out of your busy schedule to mentor a small group of Pre-cal students in Winnipeg, Manitoba. Mr. K and our class appreciate your time and sincerity. THANKS LANI!! Good day, and have fun! </span>elizabethdianecnoreply@blogger.com1tag:blogger.com,1999:blog-8985338501022987262.post-36784443760820370362007-01-28T11:31:00.000-06:002007-01-28T15:48:52.779-06:00THANK YOU Lani!<span style="color: rgb(51, 102, 102);">Thanks Lani for giving us your support. We gladly appreciate it. You've always been there cheering us on, encouraging us to move forward. I'm sure the class has a lot more to say, but nonetheless, we feel like we owe you a big thank you...<br /><br /><span style="font-weight: bold;font-size:600%;" >THANK YOU! </span></span><br /><br /><span style="color: rgb(51, 102, 102);">~ -Zeph</span><br /><br />GO PC30SF06 GO!zephnoreply@blogger.com1tag:blogger.com,1999:blog-8985338501022987262.post-3292058323217600502007-01-24T22:27:00.000-06:002007-01-24T22:52:43.958-06:00Graphing Functions NotesAs promised, here they are, the full set of notes for graphing polynomial and rational functions. These notes must be copied <b><font color="orange">by hand</font></b> into your math dictionaries. Do not hand in the printed pages. There is a good reason for this ... if you really want to know ask me in class. ;-)<br /><br />Learn Hard!<br /><br /><a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rbg2NW8CCTI/AAAAAAAAAEM/ElA2Cb-Cu_g/s1600-h/CurveSketchingNotes1.jpg">Page 1</a> of 7<br /><a href="http://bp3.blogger.com/_tlF3qs3v9cw/Rbg2V28CCUI/AAAAAAAAAEU/Up1gGKPs8dc/s1600-h/CurveSketchingNotes2.jpg">Page 2</a> of 7<br /><a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rbg2jW8CCVI/AAAAAAAAAEc/DJRQhBi7spc/s1600-h/CurveSketchingNotes3.jpg">Page 3</a> of 7<br /><a href="http://bp3.blogger.com/_tlF3qs3v9cw/Rbg2t28CCWI/AAAAAAAAAEk/aam3_ndL8_g/s1600-h/CurveSketchingNotes4.jpg">Page 4</a> of 7<br /><a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rbg24W8CCXI/AAAAAAAAAEs/VtWcYBPYgEM/s1600-h/CurveSketchingNotes5.jpg">Page 5</a> of 7<br /><a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rbg3AW8CCYI/AAAAAAAAAE0/fKdmxgouaK8/s1600-h/CurveSketchingNotes6.jpg">Page 6</a> of 7<br /><a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rbg3IW8CCZI/AAAAAAAAAE8/OcYY-ZBNiAk/s1600-h/CurveSketchingNotes7.jpg">Page 7</a> of 7Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.com5tag:blogger.com,1999:blog-8985338501022987262.post-61940686069430785782007-01-24T19:50:00.000-06:002007-01-25T11:10:19.373-06:00Scribe PostOmg sorry i got mixed up with natnael and melissas post because natnael posted after melissa but yeah i'll do monday, tuesday, and today. Once again sorry for keeping you guys waiting.<br /><br />Okay so lets start with Monday. On Monday we started off the class by dividing:<br /><br />Divide. Indicate any remainders as a fraction.<br /><br /><img src="http://img.asiantown.net/members/21921/0div.JPG"><br />So <strong>question 1</strong> had no remainders but number 2 did have a remainder of 2. <br /><br />The quotient of <strong>question 2</strong> can also be written as 71<sup>2/3</sup>. <br /><br /><strong>Question 3</strong> had no remainder but the quotient can be simplified to 2x<sup>2</sup>+5x+3=(x+1)(2x+3). <br /><br /><strong>Question 4</strong> has a remainder like question number 2. it can be written as x<sup>3</sup>-4x<sup>2</sup>+2x-1=x<sup>2</sup>-2x-2-5/x-2<br /><br /><br />Instead of doing division the long way, we learned how to do <strong><u>synthetic division</strong></u>. This is what synthetic division looks like:<br /><br /><img src="http://img.asiantown.net/members/21921/00div.JPG"><br /><br /><br /><br />On Tuesday we started off the class by finding intercepts.<br /><br /><b><u>Find all intercepts</u></b><br />f(x)=4x<sup>3</sup>-7x+3<br /><b>*The degree of the polynomial determines the maximum roots that the graph will have. Something like 4x<sup>4</sup> could have up to 4 roots.</b><br />The first thing you do is find all the possible numerators (you can do this by finding all the factors of the constant in the equation):<br />±1, ±3<br />The next thing you do is find all the possible denominators (you can do this by finding all of the factors of the leading coefficient):<br />±1,±2,±4<br />From that you can find all the <b>possible</b> roots:<br />±/1, ±3/1, ±1/2, ±3/2, ±1/4, ±3/4<br /><br />The easiest way to test if these are roots is by synthetic division:<br /><img src="http://img.asiantown.net/members/21921/000div.JPG"><br />If there is no remainder, which in this case there isn't, you take the linear factor (x-1) and add the x<sup>2</sup> and the x where they belong then multiply the 2 to get the equation of the line. <br />f(x)=(x-1)(4x<sup>2</sup>+4x-3)<br />You can factor this further:<br />f(x)=(x-1)(4x<sup>2</sup>+4x-3)<br />f(x)=(x-1)[4x<sup>2</sup>+<u>6x-2x</u>-3]<br />f(x)=(x-1)[2x(2x+3)-1(2x+3)]<br />f(x)=(x-1)(2x+3)(2x-1)<br />From here you conclude that there are roots at (1,0), (1/2,0), (-3/2,0) and the y-intercept is at 3 because if you substitue 0 in for x in the equation, you are left with 3.<br /><br />To draw the graph of this function, you must do a sign analysis. You must substitute numbers between the roots into the equation to check if they are negative or positive. The end result for this equation looks like this:<br /><img src="http://img.asiantown.net/members/21921/0000div.JPG"><br /><b>*The only way a root can change its sign is by passing through zero.</b><br /><br />The graph of this function would look something like this:<br /><img src="http://img.asiantown.net/members/21921/0000di0v.JPG"><br /><br />On Wednesday we started by sketching the graph of <b>f(x)=-2x<sup>3</sup>-7x<sup>2</sup>2x+3</b>.<br />These are the steps you take to graph this polynomial:<br /><b>1.</b>Factor everything<br /><b>2.</b>Find the y-intercept<br /><b>3.</b>Find the roots (You may use the roots theorem)<br /><b>4.</b>Do a sign analysis<br /><b>5.</b>Sketch the graph<br /><br />This polynomial is easy to work with in this form.<br />The y-intercept is the constant which in this case is 3.<br />Find the roots by finding the possible roots and use synthetic division to find a root:<br /><img src="http://img.asiantown.net/members/21921/00000div.JPG"><br />From this, you can also find points on the graph by using the possible root as the x-coordinate and the remainder as the y-coordinate. Here you have points at (1,-8), (3,-57), and a root at (-1,0)<br /><br />You can factor the polynomial to find more roots by using the linear equation x+1 and multiplying it by the equation you got by using synthetic division:<br />f(x)=(x+1)(-2x<sup>2</sup>-x+3)<br />f(x)=(x+1)(-2x<sup>2</sup>-6x+x+3)<br />f(x)=(x+1)[-2x(x+3)+1(x+3)]<br />f(x)=(x+1)(x+3)(-2x+1)<br />From this you get roots at x=-3,-1,1/2 and you can sketch the graph of the polynomial. The graph would look something like this:<br /><img src="http://img.asiantown.net/members/21921/000000div.JPG"><br /><br />Now something like y=1/x would have a vertical asymptote on the y-axis and a horizontal asymptote on the horizontal axis. The graph would look like this:<br /><img src="http://farm1.static.flickr.com/130/369048046_9cd7831bce.jpg?v=0"><br />The next scribe is <b>eeDce</b><br /><br />The steps to graphing a function like f(x)=1/x-2 would be:<br /><b>1.</b>Factor everything<br /><b>2.</b>Find the y-intercept<br /><b>3.</b>Find the roots <br /><b>4.</b>Find the vertical asymptote<br /><b>5.</b>Find the horizontal asymptote<br /><b>6.</b>Do a sign analysis<br /><b>7.</b>Sketch the graph<br /><br />f(x)=1/x-2 is already factored<br />The y-intercept would be at y=-1/2<br />There would be no roots<br />The vertical asymptote would be at x=2<br />The horizontal asymptote would be found like this:<br />lim<sub>x<FONT FACE="Symbol">®</FONT><FONT FACE="Symbol">¥</FONT></sub> 1/x-2<br />lim<sub>x<FONT FACE="Symbol">®</FONT><FONT FACE="Symbol">¥</FONT></sub> (1/x)/(x/x-2/x)<br />lim<sub>x<FONT FACE="Symbol">®</FONT><FONT FACE="Symbol">¥</FONT></sub> (1/x)/(1-2/x)<br />=0/1-0<br />The horizontal asymptote would be at y=0 on the y-axis. An easier way to find the horizontal asymptote is by taking the leading coefficient of the denominatior and divide it by the leading coefficient of the numberator but only if they have the same degree.<br />The sign analysis would look like this:<br /><img src="http://www.flickr.com/photos/43046304@N00/369059160/"><br />The graph for this would look like this:<br /><img src="http://farm1.static.flickr.com/166/369069413_aa95ab3639.jpg?v=0"><br /><br />THE NEXT SCRIBE IS <b>eeDce</b>crysta ﺕhttp://www.blogger.com/profile/14586751208806593412noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-53218221103445296462007-01-22T15:35:00.000-06:002007-01-22T15:36:53.292-06:00Factor and Remainder Theorem HomeworkTonight's homework.<br /><br /><center><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp0.blogger.com/_tlF3qs3v9cw/RbUuR28CCSI/AAAAAAAAAEA/LtjYc9OJxVk/s1600-h/FactorRemainderTheorems.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://bp0.blogger.com/_tlF3qs3v9cw/RbUuR28CCSI/AAAAAAAAAEA/LtjYc9OJxVk/s320/FactorRemainderTheorems.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5022971843741616418" /></a><br /></center>Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-18951563638619871642007-01-21T23:13:00.001-06:002007-01-23T21:47:25.978-06:00Functions<span style="font-style: italic;">Function: Functions are set(s) of rules that turn one mathematical object into another.</span><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp1.blogger.com/_PdBkpDRi3vw/RbQmGwCoicI/AAAAAAAAABI/3NsI_4MgQN0/s1600-h/func.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp1.blogger.com/_PdBkpDRi3vw/RbQmGwCoicI/AAAAAAAAABI/3NsI_4MgQN0/s400/func.bmp" alt="" id="BLOGGER_PHOTO_ID_5022681381841963458" border="0" /></a><br /><div style="text-align: center;"><span style="font-weight: bold; color: rgb(255, 0, 0);">FUNCTIONS <span style="color: rgb(204, 0, 0);">+</span> <span style="color: rgb(0, 0, 0);">RELATIONS</span></span><br /><div style="text-align: left;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp1.blogger.com/_PdBkpDRi3vw/RbQ5ywCoigI/AAAAAAAAABo/P9SMDxwqKVc/s1600-h/relation%2Bfunction.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp1.blogger.com/_PdBkpDRi3vw/RbQ5ywCoigI/AAAAAAAAABo/P9SMDxwqKVc/s400/relation%2Bfunction.bmp" alt="" id="BLOGGER_PHOTO_ID_5022703028477135362" border="0" /></a>-All functions are relations.<br />-Not all relations are functions.<br /></div><br /><u><span style="font-style: italic;">Four ways of presenting functions: [Using the function</span><span style="font-style: italic;"> </span><span style="font-style: italic;">f(x) = x + 1]</span><br /><br /></u></div><span style="font-weight: bold;">1) Verbal</span><br />...the input changes to an output after adding one unit to it.<br /><span style="font-weight: bold;">2) Graphically:</span> Graph<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp1.blogger.com/_PdBkpDRi3vw/RbQmzwCoidI/AAAAAAAAABQ/F3SCSYzRams/s1600-h/graph+x%2B1.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp1.blogger.com/_PdBkpDRi3vw/RbQmzwCoidI/AAAAAAAAABQ/F3SCSYzRams/s400/graph+x%2B1.bmp" alt="" id="BLOGGER_PHOTO_ID_5022682154936076754" border="0" /></a><br /><span style="font-weight: bold;">3) Numerically:</span> Table of Values<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp3.blogger.com/_PdBkpDRi3vw/RbQ4PQCoifI/AAAAAAAAABg/xKgSoOiBcjk/s1600-h/table+of+values.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp3.blogger.com/_PdBkpDRi3vw/RbQ4PQCoifI/AAAAAAAAABg/xKgSoOiBcjk/s400/table+of+values.bmp" alt="" id="BLOGGER_PHOTO_ID_5022701319080151538" border="0" /></a><br /><span style="font-weight: bold;">4) Symbolically: </span>Equation<br /><span style="font-style: italic;">f(x) = x + 1<br /><span style="font-style: italic;"><br /></span></span><div style="text-align: center;"><span style="font-style: italic;"><span style="font-style: italic;"><u>FLAVORS OF FUNCTIONS<br /><br /></u></span></span><div style="text-align: left;"><div style="text-align: center;"><span style="font-weight: bold;">1)Many-to-One Function</span><br /></div> This is when a function provides one output for two or more inputs.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp0.blogger.com/_PdBkpDRi3vw/RbQ8dgCoihI/AAAAAAAAABw/QcLe0Hc2Al4/s1600-h/many-to-one.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp0.blogger.com/_PdBkpDRi3vw/RbQ8dgCoihI/AAAAAAAAABw/QcLe0Hc2Al4/s400/many-to-one.bmp" alt="" id="BLOGGER_PHOTO_ID_5022705961939798546" border="0" /></a><br />in the above example; when the input is -2 the output is 3.5, and when the input is 2 the output is 3.5 again.<br /><br /><div style="text-align: center;"><span style="font-weight: bold;">2) One-to-One Function:</span><br /></div> This is a kind function, where there's is only one output for one input.<br /><span style="font-style: italic;">E.g f(x) = x - 3</span><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp3.blogger.com/_PdBkpDRi3vw/RbQ9jQCoiiI/AAAAAAAAAB4/I-JHWaLty8I/s1600-h/one-to-one.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 241px; height: 162px;" src="http://bp3.blogger.com/_PdBkpDRi3vw/RbQ9jQCoiiI/AAAAAAAAAB4/I-JHWaLty8I/s400/one-to-one.bmp" alt="" id="BLOGGER_PHOTO_ID_5022707160235674146" border="0" /></a><br /><div style="text-align: center;"><span style="font-weight: bold;">FUNCTION COMPOSITION</span><br /></div>Function composition dimply means: feeding a function into another.<br />for e.g.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp2.blogger.com/_PdBkpDRi3vw/RbRBfACoijI/AAAAAAAAACA/AlrdsY0bosc/s1600-h/function+composition.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp2.blogger.com/_PdBkpDRi3vw/RbRBfACoijI/AAAAAAAAACA/AlrdsY0bosc/s400/function+composition.bmp" alt="" id="BLOGGER_PHOTO_ID_5022711485267741234" border="0" /></a><span style="color: rgb(102, 102, 102);">Note: We can use all the operations on function compositions.<br /><br />TEST FOR FUNCTIONS:<br />This is done by the vertical line test, which ensures that there's only one output for each input.<br />e.g. 1\ Passes 2\ Fails<br /></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp0.blogger.com/_PdBkpDRi3vw/RbRHKgCoikI/AAAAAAAAACI/-DtX8XVmra4/s1600-h/fail%2Bpass.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp0.blogger.com/_PdBkpDRi3vw/RbRHKgCoikI/AAAAAAAAACI/-DtX8XVmra4/s400/fail%2Bpass.bmp" alt="" id="BLOGGER_PHOTO_ID_5022717730150189634" border="0" /></a><br /></div></div>Natnaelhttp://www.blogger.com/profile/04023224057112546632noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-13845544708478108012007-01-21T16:44:00.000-06:002007-01-21T18:05:39.299-06:00Baby play + Parent clean up = Functions?<span style="font-family:verdana;">Hello!</span><br /><br /><span style="font-family:verdana;">Friday's class Mr.K wanted to tell us a story. He told us a story about his daughter playing in a play room, where she would disorganize everything. Then Mr.k and his wife would put everything back in place afterwards.</span><br /><br /><span style="font-family:verdana;">At first most of us thought it had nothing to do with functions.</span><br /><br /><a style="font-family: verdana;" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp2.blogger.com/_tn1liJ1zVUc/RbPvhp0vyYI/AAAAAAAAAAc/Avd7EzTi-mE/s1600-h/functions.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp2.blogger.com/_tn1liJ1zVUc/RbPvhp0vyYI/AAAAAAAAAAc/Avd7EzTi-mE/s320/functions.bmp" alt="" id="BLOGGER_PHOTO_ID_5022621370890045826" border="0" /></a><br /><br /><span style="font-family:verdana;">This </span><span style="font-weight: bold; color: rgb(51, 0, 51);font-family:verdana;" >chart</span><span style="font-family:verdana;"> helped us understand more about inverse functions.</span><br /><br /><span style="font-weight: bold; color: rgb(51, 0, 51);font-family:verdana;" >Inverse Functions: </span><span style="color: rgb(102, 51, 102);font-family:verdana;" >undoes the input of the function (Exchanging the domain and range)<br /><br /><span style="color: rgb(0, 0, 0);">We then moved on to some examples.<br /><br /><span style="font-weight: bold;">f (x) =2x-3</span> <span style="color: rgb(102, 51, 102);">(inverse will be to divide. Changing the order of operations) </span><br /><span style="font-weight: bold;">y=2x-3<br /></span><br />Inverse functions of <span style="font-weight: bold;">y=2x-3.<br />x=2y-3<br />x+3=2y<br />x+3/2=y or 1/2x + 3/2=y<br /></span><br />We can really understand it more when we look at the<span style="font-weight: bold;"> table of values.<br /><br /></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp0.blogger.com/_tn1liJ1zVUc/RbP6lJ0vyZI/AAAAAAAAAAo/fInnW2r35yI/s1600-h/functions1.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp0.blogger.com/_tn1liJ1zVUc/RbP6lJ0vyZI/AAAAAAAAAAo/fInnW2r35yI/s320/functions1.bmp" alt="" id="BLOGGER_PHOTO_ID_5022633525647493522" border="0" /></a>Another example: <span style="text-decoration: underline;"><br /></span><br />f-1(x) = 3<span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);">√5x2-2/6<br /><br />x = 3</span></span></span></span></span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);">√5y2-2/6<br /><br />x3 = 5y2-2/6<br /><br />6x3 = 5y2-2<br /><br />6x3+2 = 5y2<br /><br />6x3+2 = 5y2<br /><br />6x3+2/5 = y2<br /><br /></span></span></span></span></span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);">√</span></span></span></span></span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);">6x3+2/5 = y<br /><br /></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp3.blogger.com/_tn1liJ1zVUc/RbP_R50vyaI/AAAAAAAAAAw/88lKZawbFVg/s1600-h/fusdhigahskdlj.bmp"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://bp3.blogger.com/_tn1liJ1zVUc/RbP_R50vyaI/AAAAAAAAAAw/88lKZawbFVg/s320/fusdhigahskdlj.bmp" alt="" id="BLOGGER_PHOTO_ID_5022638692493150626" border="0" /></a><br /><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);"><br /></span></span></span></span></span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(204, 0, 0);"><span style="color: rgb(0, 0, 0);"><br /></span></span></span></span></span><span style="color: rgb(102, 51, 102);font-family:verdana;" ><span style="color: rgb(0, 0, 0);"><span style="font-weight: bold;"><br /><br /><br /></span></span><br /><br /><br />The next scribe for tomorrow is </span><span style="font-weight: bold; color: rgb(51, 0, 51);">Crysta!</span><br /><span style="color: rgb(102, 51, 102);font-family:verdana;" ><br /></span>Mel!ssahttp://www.blogger.com/profile/07224180781002296901noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-82473683254628936282007-01-17T07:17:00.000-06:002007-01-17T07:26:19.186-06:00BOB - LogicsI'd have to say that this unit was quite different from the units that we previously worked on. Some parts were easier for me, so I hope I'll be able to boost up my mark through this. The same goes with everyone else. To be honest, I found it confusing at first, but since it related to real life, I began seeing the concept in a different perspective where I can understand it better. It definitely got my mind going though. So, hopefully this test will be smooth sailing for everyone. <br /><br />Good luck everyone!Mary Annhttp://www.blogger.com/profile/04859803744380122238noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-83988615988882830972007-01-17T00:58:00.000-06:002007-01-17T01:08:38.912-06:00Hey BoB .. you've got anything to say?...no? ...y?Hey guys,<br />i couldn't get to class today. well, i had a doc. appointment, and then felt like i needed rest. so, anyhow...this unit i would say was pretty interesting and sometimes mentally challenging (well, not quite).<br /> Logic + Statistics are my favourite branches of math, cuz they dont just relate to real life. They are real life lol. And in this unit, i would say i tried my best, and it went fairly good, accept that i missed the last day. anyways, I'm hopin that i didnt miss that much, and am looking forward for a good grade on this test.<br /><br />Hope this test will boost up everyones average!<br /><br />NatiNatnaelhttp://www.blogger.com/profile/04023224057112546632noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-88398229988809145142007-01-16T22:53:00.000-06:002007-01-16T23:01:22.605-06:00BOB TIMEE - LOGICMy view on our logic unit was a fair stretch of our mind. I think we may have all found it difficult only because our state of mind and way of thinking differs from dealing with triangles and square roots. Some particular parts of this unit was very plain and simple (i.e: the venn diagrams). Others required more thought and time (i.e: deductive/inductive reasoning). Although everything was possible to solve with either a formula or a broad set of rules to be followed. I hope we all do well on this test. I know its getting harder to stay consitent with our motivation to do well because the course is so close to over. I know for myself, I find myself slacking because I'm just so ready to be done. I know it's not smart on my behalf and I can already see the consequences by suffering of overload before exams. I just need to straighten out and prepare myself with a logical matter! Okay. GOOD LUCK GUYS<br /><br />I had a fun time BOBing. Hey guys its our last one!elizabethdianecnoreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-65441392418854547492007-01-16T22:09:00.000-06:002007-01-16T22:15:01.367-06:00Blogging on blobbingI personally think that this logic unit was somekind of a very...very ... long stretch for my mind. I think that i struggle for the most part of this unit. Perhaps , the easiest thing in this unit was the venn diagrams and valid and invalid arguments. I think that I really have to study very hard tonight or else i'm done. Well thats it I hope that I didnt miss too much information from todays class. Study hard and keep your body away from sickness!! Good nightm@rkhttp://www.blogger.com/profile/00509544958003356512noreply@blogger.com1tag:blogger.com,1999:blog-8985338501022987262.post-56676739052773022872007-01-16T21:03:00.000-06:002007-01-16T22:09:30.574-06:00Direct and Indirect reasoningGood evening everyonethis is m@rk scribing for yesterday's class. I've been banged up by high fever that is why i was unable to go to class today.<br /><br />To start yesterday's class we proved using direct reasoning that the product of 2 even numbers is even too.First of all an even number is something that is divisible by 2 and has a factor of 2.<br /><br />Let 2x=an even number<br />2a=another even number<br /><br />2x * 2a =4ax<br />2x * 2a= 2 (2ax)<br /><br />∴ Any product by both even numbers is also even as seen from above.<br /><br />Then, we proved using direct reasoning that the product of 2 odd numbers is odd too.<br /><br />Let (2x+1) =an odd number<br />(2a+1) = another odd number<br /><br />(2x+1) (2a+1) =4ax+2x+2a+1<br />(2x+1) (2a+1)=2 (2ax+x+a) 1<br /><br />∴ Any product by both odd numbers is also odd as seen from above because the product does not factor perfectly with 2.<br /><br />Now, from the previous class we proved that √2 is irrational using indirect reasoning.<br /><br />First, assume that √2 is rational.<br /><br />√2=a/b<br /><br />Note: a/b is in reduced form meaning they have no common factors.<br /><br />2= a²/b² <span style="color: rgb(0, 102, 0);">Multiply both sides by b²<br /><span style="color: rgb(0, 0, 0);">2b² =a²<br /><br /><span style="color: rgb(0, 102, 0);">a² must be an even number and a must be even too so, let a=2c.<br /><br /><span style="color: rgb(0, 0, 0);">2b² = (2c)²</span><br /><span style="color: rgb(0, 0, 0);">2b²=4c²</span><br /><span style="color: rgb(0, 0, 0);">b²=2c²</span><br /><br />b² and b are both even numbers so, let b=2d. <span style="color: rgb(204, 0, 0);">This is a contradiction! Both a and b cannot have any common factors because of our assumption that they are in reduced forms in the beginning.<br /><br />∴ √2 is irrational<br /><br /><span style="color: rgb(0, 0, 0);">Then, we prooved that √3 is irrational using indirect reasoning.<br /><br />First, assume that √3 is rational. THis problem is similar to the the previous one.<br /><br />√3= a/b<br /><br />Assume that a/b is in reduced form.<br /><br />3= a²/b²<br /><br />3b² = a²<br /><br />On the other hand lets look at square numbers first.<br />4= 2*2<br />9=3*3<br />16=2*2*2*2<br />25=5*5<br />36=2*3*2*3<br />49=7*7<br />64=2*2*2*2*2*2<br />81=3*3*3*3<br />100=5*2*5*2<br /><br /><span style="color: rgb(204, 0, 0); font-weight: bold;">Can you see the pattern that there is always a pair of prime numbers in square numbers?</span><br /><br /><span style="color: rgb(0, 102, 0);">So let b</span>²=3c<br /><br />3(3c)² =a ²<br /><br />3* 9c² =a²<br /><br /><span style="color: rgb(153, 0, 0); font-weight: bold;">BUt it cannot happen because a and b will share the same factor which is 3. Therefore contradicting with our first statement that a and b is in reduced form.<br /><br />∴√3 is irrational<br /><br /></span><span style="color: rgb(153, 0, 0);"><span style="color: rgb(0, 0, 0);">Lastly, we talked about pardoxes. <span style="font-weight: bold;">A paradox is a statement that cannot be true but also cannot be false.<br /><br />Example:<br />This sentence is false<br /><br /></span>In this case you cannot decide whether it is true or false.<br /><br />Thats it for my scribe for tonight. I dont know if im going to be able to go to class tomorrow. It all depends if im feeling better . LIke i told Mr. K the scribe will be Natnael. The homework is the Unusual school play<br /><span style="font-weight: bold;"></span></span></span></span></span></span></span></span>m@rkhttp://www.blogger.com/profile/00509544958003356512noreply@blogger.com2tag:blogger.com,1999:blog-8985338501022987262.post-44448803094248509812007-01-16T20:25:00.000-06:002007-01-16T20:28:21.495-06:00BOBAnother unit test coming up. For me, logic was a lot harder than I thought it would be. Even if it's not that easy, I personally enjoy the challenge of finding the solutions. I've had experience with these kinds of problems in junior high, so I have a good starting idea as to how to approach the question. I just hope that we all do really good on this test, and hopefully, I get the mark I've worked to get. 'Night!cherrieeeeeeeehttp://www.blogger.com/profile/07176149235231149728noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-49975062766871801182007-01-16T19:33:00.000-06:002007-01-16T19:40:56.333-06:00BOBThis logic unit was interesting. It was cool getting back to the Venn-diagrams, bringing back the memories of arithmetic's in elementary. The good old days, everything was so simple. This unit got more difficult when we started the inductive and deductive reasoning. I'm still having trouble with that. <br /><br />That's all, good luck to everyone tomorrow!Mel!ssahttp://www.blogger.com/profile/07224180781002296901noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-36634637831702259842007-01-16T07:04:00.000-06:002007-01-16T22:41:40.165-06:00BOB v.6 The LOGICal BobWell I can't say the logic unit was straight forward since we now know about the paradoxes. It was fun though doing those paradox assignments on "A Usual Day at an Unusual School." Venn Diagrams were easy to do and so were the matrix problems. It felt like doing jr.high stuff but more advance. It was awesome learning logic in math class where we discuss things that barely have to do with numbers as compared to the other unit we're doing - consumer math. We discussed confusing things suchas <span style="font-style: italic;">"the inverse is the converse of the contrapositive"</span> and the existence of counterexamples. Learning logic was fun and very interesting and the countdown begins for the logic test to show.<br /><br />-Zephzephnoreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-86746523725785166202007-01-15T21:49:00.000-06:002007-01-15T22:35:11.017-06:00BOBing<span style="color: rgb(51, 153, 153);">This unit was alright, logic. It was difficult at first because I wasn't sure how to start solving the problems. I like doing problems where you figure out if it's a deductive information or inductive information. I also like doing problems where you verify if it's a valid argument or invalid argument.</span><br /><span style="color: rgb(51, 153, 153);">What I found most difficult is when I try to figure out if the problem is a paradox or not. Proving that the square root of some numbers are irrational is also something that I find difficult as well. The consumer math stuff was alright. There were times where I didn't understand it, that was when the question asked about stuff I didn't know yet. My parents helped me, because they deal with this stuff all year round. Overall, consumer math was good and logic was quite difficult. Well, I'll see you all in class tomorrow fellow classmates and I hope you all study hard! </span><br /><br /><span style="color: rgb(51, 153, 153);">-<span style="font-style: italic;"><span onclick="BLOG_clickHandler(this)" class="blsp-spelling-error" id="SPELLING_ERROR_0">SAMUS</span> :D</span></span>SAMUSnoreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-86506404748138296972007-01-15T21:36:00.000-06:002007-01-15T21:43:20.051-06:00BOB for Logics test in 1day and 16 hoursIn logic we had done logical work like indirect and direct reasoning, inductive and deductive reasoning and that is what I am having trouble in. I don't understand the difference between the 2 reasoning in each set. I don't understand the definition on inductive and deductive reasoning.<br /> Something that I am getting the hang of the is the Venn-diagram questions. They seem to be the easiest parts of logic.<br /> In conclusion good luck everybody on the test that is happening in 1 day and 15 hours and 15 minutes from when I post this BOB :-)<br /><br />Signed, <br />Benvolio (Ben)Benofschoolhttp://www.blogger.com/profile/10547772016514296237noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-8527642951978173712007-01-15T21:13:00.000-06:002007-01-15T21:25:45.412-06:00blogging on blogging - logicWell, this unit has gone by pretty fast. Of course that's not including the two weeks break we had. So not including that, it hasn't really been that many classes. Logic has been one of the more interesting, fun topics. Venn Diagrams, once you understand the method of how you put them together, they're pretty straight forward. And personally, if i know how to do something, it always makes it fun because i understand it. As for inductive and deductive, that's okay. i had some trouble at first, but nothing i couldn't fix. And what we did in class today was fun as well as our homework. So this unit isn't too bad.<br /><br />Honestly, i can't remember if Mr. k said to do a blog for this unit or not? A lot of things lately have just been a dream, haha, so I'm not too sure. But better to do this than not to! Well, good luck everyone.honey♥bunchhttp://www.blogger.com/profile/06630005400121287023noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-55333770603592168362007-01-15T15:45:00.000-06:002007-01-15T16:16:13.160-06:00Usual Day At Unusual SchoolWe talked about proof by contradiction in the last couple of classes. Today we proved that √<span style="text-decoration: overline">2</span> is irrational using proof by contradiction (which is also called, in latin, <a href="http://www.mathacademy.com/pr/prime/articles/irr2/index.asp">reductio ad absurdum</a>). You can also review direct proof <a href="http://www.sparknotes.com/math/geometry3/geometricproofs/section2.rhtml">here</a> (use the pop-up menu to review indirect proof and other topics as well).<br /><br />Tonight's homework is the play below. Make a list of all the Braves and Brights. We'll discuss it in class tomorrow. ;-)<br /><br /><a href="http://bp3.blogger.com/_tlF3qs3v9cw/Rav7465fA6I/AAAAAAAAACg/XvzfrOjA-3s/s1600-h/UsualDayUnusualSchool1.jpg">Page 1</a> of 4 <a href="http://bp2.blogger.com/_tlF3qs3v9cw/Rav8iq5fA7I/AAAAAAAAACs/2hKzhvhiBTY/s1600-h/UsualDayUnusualSchool2.jpg">Page 2</a> of 4 <a href="http://bp2.blogger.com/_tlF3qs3v9cw/Rav8rq5fA8I/AAAAAAAAAC0/94P0wcKQvU4/s1600-h/UsualDayUnusualSchool3.jpg">Page 3</a> of 4 <a href="http://bp1.blogger.com/_tlF3qs3v9cw/Rav80a5fA9I/AAAAAAAAAC8/AsQV5gHdjcw/s1600-h/UsualDayUnusualSchool4.jpg">Page 4</a> of 4Darren Kuropatwahttps://plus.google.com/116565501620976083429noreply@blogger.com0tag:blogger.com,1999:blog-8985338501022987262.post-42094062483568065282007-01-15T07:47:00.000-06:002007-01-15T07:49:34.234-06:00Challenge<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://home.alltel.net/lanihall/challenge.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px;" src="http://home.alltel.net/lanihall/challenge.jpg" border="0" alt="" /></a><br /><p>It's a New Year, with new tests, new units, new learning! And how will you meet those challenges?Lani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com0